146 research outputs found
Non-Separable, Quasiconcave Utilities are Easy -- in a Perfect Price Discrimination Market Model
Recent results, establishing evidence of intractability for such restrictive
utility functions as additively separable, piecewise-linear and concave, under
both Fisher and Arrow-Debreu market models, have prompted the question of
whether we have failed to capture some essential elements of real markets,
which seem to do a good job of finding prices that maintain parity between
supply and demand.
The main point of this paper is to show that even non-separable, quasiconcave
utility functions can be handled efficiently in a suitably chosen, though
natural, realistic and useful, market model; our model allows for perfect price
discrimination. Our model supports unique equilibrium prices and, for the
restriction to concave utilities, satisfies both welfare theorems
An Incentive Compatible, Efficient Market for Air Traffic Flow Management
We present a market-based approach to the Air Traffic Flow Management (ATFM)
problem. The goods in our market are delays and buyers are airline companies;
the latter pay money to the FAA to buy away the desired amount of delay on a
per flight basis. We give a notion of equilibrium for this market and an LP
whose solution gives an equilibrium allocation of flights to landing slots as
well as equilibrium prices for the landing slots. Via a reduction to matching,
we show that this equilibrium can be computed combinatorially in strongly
polynomial time. Moreover, there is a special set of equilibrium prices, which
can be computed easily, that is identical to the VCG solution, and therefore
the market is incentive compatible in dominant strategy.Comment: arXiv admin note: substantial text overlap with arXiv:1109.521
Matching Is as Easy as the Decision Problem, in the NC Model
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for
it? This has been an outstanding open question in TCS for over three decades,
ever since the discovery of randomized NC matching algorithms [KUW85, MVV87].
Over the last five years, the theoretical computer science community has
launched a relentless attack on this question, leading to the discovery of
several powerful ideas. We give what appears to be the culmination of this line
of work: An NC algorithm for finding a minimum-weight perfect matching in a
general graph with polynomially bounded edge weights, provided it is given an
oracle for the decision problem. Consequently, for settling the main open
problem, it suffices to obtain an NC algorithm for the decision problem. We
believe this new fact has qualitatively changed the nature of this open
problem.
All known efficient matching algorithms for general graphs follow one of two
approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based
algorithm follows a new approach and uses many of the ideas discovered in the
last five years.
The difficulty of obtaining an NC perfect matching algorithm led researchers
to study matching vis-a-vis clever relaxations of the class NC. In this vein,
recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC
algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC
algorithm with the additional requirement that on the same graph, it should
return the same (i.e., unique) perfect matching for almost all choices of
random bits. A corollary of our reduction is an analogous algorithm for general
graphs.Comment: Appeared in ITCS 202
On Computability of Equilibria in Markets with Production
Although production is an integral part of the Arrow-Debreu market model,
most of the work in theoretical computer science has so far concentrated on
markets without production, i.e., the exchange economy. This paper takes a
significant step towards understanding computational aspects of markets with
production.
We first define the notion of separable, piecewise-linear concave (SPLC)
production by analogy with SPLC utility functions. We then obtain a linear
complementarity problem (LCP) formulation that captures exactly the set of
equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production,
and we give a complementary pivot algorithm for finding an equilibrium. This
settles a question asked by Eaves in 1975 of extending his complementary pivot
algorithm to markets with production.
Since this is a path-following algorithm, we obtain a proof of membership of
this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of
existence of equilibrium (i.e., without using a fixed point theorem),
rationality, and oddness of the number of equilibria. We further give a proof
of PPAD-hardness for this problem and also for its restriction to markets with
linear utilities and SPLC production. Experiments show that our algorithm runs
fast on randomly chosen examples, and unlike previous approaches, it does not
suffer from issues of numerical instability. Additionally, it is strongly
polynomial when the number of goods or the number of agents and firms is
constant. This extends the result of Devanur and Kannan (2008) to markets with
production.
Finally, we show that an LCP-based approach cannot be extended to PLC
(non-separable) production, by constructing an example which has only
irrational equilibria.Comment: An extended abstract will appear in SODA 201
NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs
In 1988, Vazirani gave an NC algorithm for computing the number of perfect
matchings in -minor-free graphs by building on Kasteleyn's scheme for
planar graphs, and stated that this "opens up the possibility of obtaining an
NC algorithm for finding a perfect matching in -free graphs." In this
paper, we finally settle this 30-year-old open problem. Building on recent NC
algorithms for planar and bounded-genus perfect matching by Anari and Vazirani
and later by Sankowski, we obtain NC algorithms for perfect matching in any
minor-closed graph family that forbids a one-crossing graph. This family
includes several well-studied graph families including the -minor-free
graphs and -minor-free graphs. Graphs in these families not only have
unbounded genus, but can have genus as high as . Our method applies as
well to several other problems related to perfect matching. In particular, we
obtain NC algorithms for the following problems in any family of graphs (or
networks) with a one-crossing forbidden minor:
Determining whether a given graph has a perfect matching and if so,
finding one.
Finding a minimum weight perfect matching in the graph, assuming
that the edge weights are polynomially bounded.
Finding a maximum -flow in the network, with arbitrary
capacities.
The main new idea enabling our results is the definition and use of
matching-mimicking networks, small replacement networks that behave the same,
with respect to matching problems involving a fixed set of terminals, as the
larger network they replace.Comment: 21 pages, 6 figure
A Market for Air Traffic Flow Management
The two somewhat conflicting requirements of efficiency and fairness make
ATFM an unsatisfactorily solved problem, despite its overwhelming importance.
In this paper, we present an economics motivated solution that is based on the
notion of a free market. Our contention is that in fact the airlines themselves
are the best judge of how to achieve efficiency and our market-based solution
gives them the ability to pay, at the going rate, to buy away the desired
amount of delay on a per flight basis.
The issue of fairness is simply finessed away by our solution -- whoever pays
gets smaller delays. We show how our solution has the potential of enabling
travelers from a large spectrum of affordability and punctuality requirements
to achieve an end that is most desirable to them.
Our market model is particularly simple, requiring only one parameter per
flight from the airline company. Furthermore, we show that it admits a
combinatorial, strongly polynomial algorithm for computing an equilibrium
landing schedule and prices
The Investment Management Game: Extending the Scope of the Notion of Core
The core is a dominant solution concept in economics and cooperative game
theory; it is predominantly used for profit, equivalently cost or utility,
sharing. This paper demonstrates the versatility of this notion by proposing a
completely different use: in a so-called investment management game, which is a
game against nature rather than a cooperative game. This game has only one
agent whose strategy set is all possible ways of distributing her money among
investment firms. The agent wants to pick a strategy such that in each of
exponentially many future scenarios, sufficient money is available in the right
firms so she can buy an optimal investment for that scenario. Such a strategy
constitutes a core imputation under a broad interpretation, though traditional
formal framework, of the core. Our game is defined on perfect graphs, since the
maximum stable set problem can be solved in polynomial time for such graphs. We
completely characterize the core of this game, analogous to Shapley and Shubik
characterization of the core of the assignment game. A key difference is the
following technical novelty: whereas their characterization follows from total
unimodularity, ours follows from total dual integralityComment: 16 pages. arXiv admin note: text overlap with arXiv:2209.0490
LP-Duality Theory and the Cores of Games
LP-duality theory has played a central role in the study of the core, right
from its early days to the present time. The 1971 paper of Shapley and Shubik,
which gave a characterization of the core of the assignment game, has been a
paradigm-setting work in this regard. However, despite extensive follow-up
work, basic gaps still remain. We address these gaps using the following
building blocks from LP-duality theory:
1). Total unimodularity (TUM).
2). Complementary slackness conditions and strict complementarity.
TUM plays a vital role in the Shapley-Shubik theorem. We define several
generalizations of the assignment game whose LP-formulations admit TUM; using
the latter, we characterize their cores. The Hoffman-Kruskal game is the most
general of these. Its applications include matching students to schools and
medical residents to hospitals, and its core imputations provide a way of
enforcing constraints arising naturally in these applications: encouraging
diversity and discouraging over-representation.
Complementarity enables us to prove new properties of core imputations of the
assignment game and its generalizations.Comment: 30 pages. arXiv admin note: text overlap with arXiv:2202.0061
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